Before starting to discuss geometry, I would like to mention a few resources that I have found invaluable in my studies. When I was first studying geometry in high school in 1965, I found a great book in our public library by Nathan Altshiller-Court called College Geometry, published in 1925. I didn't understand very much of it then, but I enjoyed skimming through it and was amazed at the complexity of something as simple as the triangle. In 1975, with my BSChE, and living in Boston, I found a 1969 reprinted edition in a Harvard Square bookstore. It has been one of my favorite references ever since. There is currently a new reprint that is available on Amazon.com for a very reasonable price.
While working in Tennessee in 1970 as a co-op student at the Oak Ridge National Laboratory, I found in the Lab's library a paperback book by Roger A. Johnson called Advanced Euclidean Geometry. It was a 1960 reprint of the original 1929 work that was called Modern Geometry. I eventually found a used copy of the 1929 edition to add to my math library. There is also a new affordable reprint available on Amazon.com.
I highly recommend either of these books for anyone seriously interested in the geometry of the triangle. Or, better still, get them both, since although there is quite a bit of overlap in material, they both have some unique topics. Each covers elementary topics such as medians, altitudes and angle bisectors, and then delves deeper into the geometry of Lemoine and Brocard. Court gives many exercises for the student (without solutions), whereas, Johnson leaves the proof of some theorems to the student. Of the thirty or more books in my geometry library, these two stand above all the rest in their thorough coverage of the modern Euclidean geometry of the triangle.
When most people think of the year 1968, they remember radical students wearing love beads and peace symbols and prote! sting the Vietnam War. On the other hand, I was roaming the ha! lls of C ardinal High School in northeast Ohio with my ruler and compass in my shirt pocket, ready to construct a drawing to illustrate any geometric idea that might pop into my head. In fact, the ruler and compass remained my tools for studying geometry until 1983 when I bought a Commodore 64 computer and wrote a program that would allow me to make, save and print geometric drawings quickly and very accurately. Today there is a wide range of excellent dynamic geometry software available. I will mention the two that I use most often. First, there is The Geometer's Sketchpad (GSP) by Key Curriculum Press. This program acts just like a ruler and compass in that you can draw lines and circles and find points of intersection. There are also a few major features that make it unlike the ruler and compass. The accuracy is what you would expect from any computer calculation, with measurements given to one one-hundredth of a uni! t. The custom toolbox feature lets you construct objects and then save them as tools to be used later. For instance, if you start with three points, draw a triangle, find its orthocenter and save it as a tool, then when you use the tool you simply click on any three points and the tool draws the triangle and its orthocenter. The dynamic feature refers to the fact that you can drag any point and all other structures that are derived from that point will also move to keep their same relationship. For example, if you draw a triangle and its circumcircle and then drag one of the vertices, the circle will automatically change to remain the circumcircle of the new triangle. The program has all the typical features of a Windows program: edit, save, print, etc. My copy of the program cost about $40 a few years ago.
Another software program I have recently begun to use is C.a.R! . which is available free on the Internet from its de! veloper Rene Grothmann. This program is similar to GSP, with a couple of unique features. First, it will draw conic sections (although they are not needed in the study of the Euclidean geometry of the triangle). Also, you can set up construction problems called "Assignments". As an example, say you drew a triangle and its orthocenter and then hide everything but the three vertices and the orthocenter. To make an assignment you would target one or more points to find, say one of the vertices. Then the program would show the two other vertices and the orthocenter and let you construct the missing vertex. When you had a correct solution the program would tell you. This program is a wonderful tool for exploring the triangle and setting up problems to solve. I will have a lot more to say about it in future posts and I highly recommend it to everyone.
college geometry constructions
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